Python: fitting matrix equation with shared parameters

Question

I have two equations that originate from one single matrix equation:

[x,y] = [[cos(n), -sin(n)],[sin(n), cos(n)]]*[x', y'],

where x' = Acos(w1*t+ p1) and y' = Bcos(w2*t + p2).

This is just a single matrix equation for the vector [x,y], but it can be decomposed into 2 scalar equations: x = A*cos(s)*cos(w1*t+ p1)*x' - B*sin(s)*sin(w2*t + p2)*y' and y = A*sin(s)*cos(w1*t+ p1)*x' + B*cos(s)*sin(w2*t + p2)*y'.

So I am fitting two datasets, x vs. t and y vs. t, but there are some shared parameters in these fits, namely A, B and s.

1) Can I fit directly the matrix equation, or do I have to decompose it into scalar ones? The former would be more elegant.

2) Can I fit with shared parameters on curve_fit? All relevant questions use other packages.

your observation that all relevant question use other packages may be an interesting clue that `curve_fit` cannot do this as easily as other packages can. It may be that those other packages exist precisely because of such limitations. — M Newville, Jun 21 '18 at 22:40

score 1 · Accepted Answer · answered Jun 19 '18 at 21:15

The below example code fits two different equations with one shared parameter using curve_fit. This is not an answer to your question, but I cannot format code in a comment so I'm posting it here.

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

y1 = np.array([ 16.00,  18.42,  20.84,  23.26,  25.68])
y2 = np.array([-20.00, -25.50, -31.00, -36.50, -42.00])
comboY = np.append(y1, y2)

h = np.array([5.0, 6.1, 7.2, 8.3, 9.4])
comboX = np.append(h, h)


def mod1(data, a, b, c): # not all parameters are used here
        return a * data + c


def mod2(data, a, b, c): # not all parameters are used here
        return b * data + c


def comboFunc(comboData, a, b, c):
    # single data set passed in, extract separate data
    extract1 = comboData[:len(y1)] # first data
    extract2 = comboData[len(y2):] # second data

    result1 = mod1(extract1, a, b, c)
    result2 = mod2(extract2, a, b, c)

    return np.append(result1, result2)


# some initial parameter values
initialParameters = np.array([1.0, 1.0, 1.0])

# curve fit the combined data to the combined function
fittedParameters, pcov = curve_fit(comboFunc, comboX, comboY, initialParameters)

# values for display of fitted function
a, b, c = fittedParameters

y_fit_1 = mod1(h, a, b, c) # first data set, first equation
y_fit_2 = mod2(h, a, b, c) # second data set, second equation

plt.plot(comboX, comboY, 'D') # plot the raw data
plt.plot(h, y_fit_1) # plot the equation using the fitted parameters
plt.plot(h, y_fit_2) # plot the equation using the fitted parameters
plt.show()

print(fittedParameters)

Python: fitting matrix equation with shared parameters

1 Answers1